Dynamical Scaling and Phase Coexistence in Topologically Constrained DNA Melting

There is a long-standing experimental observation that the melting of topologically constrained DNA, such as circular closed plasmids, is less abrupt than that of linear molecules. This finding points to an important role of topology in the physics of DNA denaturation, which is, however, poorly understood. Here, we shed light on this issue by combining large-scale Brownian dynamics simulations with an analytically solvable phenomenological Landau mean field theory. We find that the competition between melting and supercoiling leads to phase coexistence of denatured and intact phases at the single-molecule level. This coexistence occurs in a wide temperature range, thereby accounting for the broadening of the transition. Finally, our simulations show an intriguing topology-dependent scaling law governing the growth of denaturation bubbles in supercoiled plasmids, which can be understood within the proposed mean field theory.

Figure 1

Melting curves. (a) The melting curves (fraction of denatured bp, $⟨\vartheta ⟩$) for nicked and intact polyoma DNA as a function of $T-T_c$ (data from [7]). (b) The melting curves obtained here via BD simulations of tfDNA and tcDNA molecules, with length 1000 bp and different levels of supercoiling, as a function of the (shifted) effective hydrogen bond strength ${\epsilon }_{\mathrm{HB}}-{\epsilon }_{\mathrm{HB},\mathrm{c}}$ (averaged over 5 replicas and $10^6$ BD time steps). In both experiments (a) and simulations (b), the transition appears smoother for tcDNA and the relative broadening ${\mathrm{\Delta }\tau |}_{\text{tc}}/{\mathrm{\Delta }\tau |}_{\text{tf}}\sim 3$ is in quantitative agreement. The critical bond energies for which half of the base pairs melt are ${\epsilon }_{\mathrm{HB},\mathrm{c}}/k_{B}T=1.35$ for linear DNA and 0.309, 0.238, 0.168 for ${\sigma }_0=-0.06$, 0, and 0.06, respectively. These values show that the critical bond energy decreases (linearly) with supercoiling. (c)–(f) Snapshots of typical configurations for ${\epsilon }_{\mathrm{HB}}=0.3$ $k_{B}T$ for tf (linear) and tcDNA with ${\sigma }_0=0.06$, 0, $-0.06$, respectively. Stably denatured bubbles localize at regions of high curvature (tips of plectonemes [31], highlighted by circles). In (c) the linear DNA molecule is fully denatured.

Figure 2

Phase diagram. The thick solid line represents the hidden first-order transition line ${\sigma }_c(a)$. The line-shadowed area highlights the region of absolute instability of the uniform phase; the spinodal region is colored in grey. Binodal lines are denoted as ${\sigma }_{-}$ and ${\sigma }_{+}$. The cross-shadowed area highlights the region of coexistence of two denatured (ss) phases. Filled symbols denote the values obtained from numerical integration of Eq. (4), with initial ${\sigma }_0$ as indicated by the empty squares. Snapshots of ssDNA, dsDNA, and ds-ssDNA coexistence observed in BD simulations are also shown.

Figure 3

Kymographs. Panels (a) and (b) report results from BD simulations. At time $t=0$, the H bond strength is quenched to ${\epsilon }_{\mathrm{HB}}=0.3k_{B}T$ and the local state of the chain (red for denatured and white for intact) is recorded as a kymograph. Panels (a) and (b) show the case of a tfDNA and tcDNA (${\sigma }_0=0$), respectively. Panel (c) shows the kymograph of the system during integration of Eqs. (4) starting from a small bubble (see [25]). Insets show instantaneous profiles of denaturation field (red) and supercoiling field (blue).

Figure 4

Dynamical scaling. Panels (a) and (b) show results from BD simulations. The size of the largest denatured bubble $⟨l⟩$ (averaged over five replicas) is plotted against time from the moment of the quench. Panel (a) shows tcDNA while panel (b) refers to tfDNA. Panel (c) shows the size of a single growing bubble, $l$, within our mean field model, Eqs. (4). PDE and BD simulations show similar behaviors, which suggest a universal dynamical scaling with topology-dependent exponent ($\alpha =1$ for $\chi =0$ and $\alpha =1/2$ for $\chi >0$). Panel (d) shows the linking number, ${\mathrm{Lk}}_d$, stored inside a denatured bubble of fixed size $l$ computed from BD simulations (see text and [25] for details).